Fixed-point property

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

Definition

A topological space is said to have the fixed-point property if every continuous map from the topological space to itself, has a fixed point.

Relation with other properties

Stronger properties

• acyclic compact polyhedron (nonempty)

Facts

• Lefschetz fixed-point theorem
• Brouwer fixed-point theorem

In general, we combine the Lefschetz fixed-point theorem with the structure of the cohomology ring of the space to determine whether or not it has the fixed-point property. For instance, we can show that complex projective space in even dimensions has the fixed-point property, by combining the Lefschetz fixed-point theorem with the fact that the trace on the $(2k)^{th}$ homology is $d^{k}$ where $d$ is the trace on the second homology.

Metaproperties

Retract-hereditariness

This property of topological spaces is hereditary on retracts, viz if a space has the property, so does any retract of it
View all retract-hereditary properties of topological spaces

Every retract of a space with the fixed-point property also has the fixed-point property